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  4. displaystyle lim n arrow ∞ (5n+1+3n-22 n/5n+2n+32 n+3) is equal to

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Q. $\displaystyle\lim _{n \rightarrow \infty} \frac{5^{n+1}+3^{n}-2^{2 n}}{5^{n}+2^{n}+3^{2 n+3}}$ is equal to

Limits and Derivatives

Solution:

$\displaystyle\lim _{n \rightarrow \infty} \frac{5^{n+1}+3^{n}-2^{2 n}}{5^{n}+2^{n}+3^{2 n+3}}$
$=\displaystyle\lim _{n \rightarrow \infty} \frac{5 \cdot 5^{n}+3^{n}-4^{n}}{5^{n}+2^{n}+27 \cdot 9^{n}}$
$=\displaystyle\lim _{n \rightarrow \infty} \frac{5 \cdot \frac{5^{n}}{9^{n}}+\frac{3^{n}}{9^{n}}-\frac{4^{n}}{9^{n}}}{\frac{5^{n}}{9^{n}}+\frac{2^{n}}{9^{n}}+27}$
$=\frac{0+0-0}{0+0+27}=0$